A350176 -id:A350176 - cp's OEIS frontend

A350402 Numbers k such that binomial(k, 2) divides binomial(2^k-2, 2).

2, 3, 7, 11, 19, 31, 43, 127, 163, 211, 271, 311, 331, 379, 487, 571, 631, 811, 883, 991, 1459, 1471, 1747, 2311, 2531, 2647, 2791, 2971, 3079, 3631, 3943, 4091, 5171, 5419, 6571, 7591, 8863, 8911, 9199, 9791, 9931, 10891, 11827, 11971, 13591, 14407, 15391, 16759, 17011, 18523, 19531, 21871, 22111, 23431, 24967

Offset: 1

Juri-Stepan Gerasimov, Dec 29 2021

nonn

Conjecture: aside from the first term, this is a subsequence of A094179 (numbers congruent to 3 mod 4 which are divisible only by primes congruent to 3 mod 4).

The conjecture is false: a(2295) = 508606771 = 19531 * 26041 is not in A094179, nor even A004614. - Charles R Greathouse IV, Jan 22 2022

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Supersequence of A069051.

Cf. A069051 (binomial(k,2) divides binomial(2^k-1, 2)?), A094179, A350176.

[n: n in [2..25000] |  IsZero(Binomial(2^n-2, 2) mod Binomial(n, 2))];

Select[Range[2, 25000], Divisible[Binomial[2^# - 2, 2], Binomial[#,2]] &] (* Amiram Eldar, Dec 29 2021 *)

isok(n) = (n>1) && ((binomial(2^n-2, 2) % binomial(n, 2)) == 0); \\ Michel Marcus, Jan 04 2022

is(n)=my(m=n^2-n,t=Mod(2,m)^n-2); t*(t-1)==0 \\ Charles R Greathouse IV, Jan 20 2022

A350905 Numbers k such that binomial(k, 3) divides binomial(3^k-3, 2).

Original entry on oeis.org

3, 5, 17, 37, 79, 101, 257, 14401, 44101, 47881, 57601, 65537, 677041, 1354081, 2031121, 3766141, 7812169, 8122501, 17907121, 18671941, 19676161, 21381361, 29615041, 30115009, 65246161, 105557761, 144406081, 181254529, 217039681, 242235841, 263062801, 277032001

Offset: 1

Juri-Stepan Gerasimov, Jan 21 2022

Are all terms prime numbers?

Conjecture: all terms of the intersection with A350176 are prime numbers.

Chai Wah Wu, Table of n, a(n) for n = 1..86

Supersequence of A019434.

Cf. A350176, A350402.

[n: n in [3..10^4] |  IsZero(Binomial(3^n-3, 2) mod Binomial(n, 3))];

Select[Range[3, 70000], Divisible[Binomial[3^# - 3, 2], Binomial[#, 3]] &] (* Amiram Eldar, Jan 21 2022 *)

is(k) = if(k>2, my(m=Mod(3, (k^3+2*k)/3-k^2)^k);  m^2-7*m+12==0); \\ Jinyuan Wang, Jan 22 2022

More terms from Jinyuan Wang, Jan 22 2022

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A350402 Numbers k such that binomial(k, 2) divides binomial(2^k-2, 2).

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A350905 Numbers k such that binomial(k, 3) divides binomial(3^k-3, 2).

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